Properties

Label 133200.dz
Number of curves $6$
Conductor $133200$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("133200.dz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 133200.dz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
133200.dz1 133200cs3 [0, 0, 0, -1227618075, 16555527490250] [2] 31850496  
133200.dz2 133200cs6 [0, 0, 0, -1091250075, -13814727277750] [2] 63700992  
133200.dz3 133200cs4 [0, 0, 0, -105570075, 46890562250] [2, 2] 31850496  
133200.dz4 133200cs2 [0, 0, 0, -76770075, 258368962250] [2, 2] 15925248  
133200.dz5 133200cs1 [0, 0, 0, -3042075, 7030210250] [2] 7962624 \(\Gamma_0(N)\)-optimal
133200.dz6 133200cs5 [0, 0, 0, 419309925, 373890802250] [2] 63700992  

Rank

sage: E.rank()
 

The elliptic curves in class 133200.dz have rank \(1\).

Modular form 133200.2.a.dz

sage: E.q_eigenform(10)
 
\( q + 4q^{11} + 2q^{13} + 2q^{17} - 4q^{19} + O(q^{20}) \)

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.